Vivek Hallegere Murthy's nomination is opposed by the National Rifle Association (NRA) and would almost certainly fail if brought up for a vote right now. Some reports say up to 10 Democrats could oppose the nomination.

"I think it is interesting that Harry Reid hasn't brought him up yet," Rubio told The Washington Times in a radio interview. "He is always complaining about Republicans holding up nominees, well here you go. I mean, why don't you bring him up on Monday when we get back."

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Rubio acknowledged Reid is unlikely to take him up on the offer.

"But they are not doing that because they don't have the votes,” Rubio said. “But he will go down. And he will not have support to pass the Senate. Remember a handful of Democrats are prepared to vote against him.”

The White House has said it is "recalibrating" its strategy on moving forward with the nomination. Nonetheless, press secretary Jay Carney said Murthy would be a powerful messenger on health issues if confirmed.

“Dr. Murthy is a dynamic, entrepreneurial practitioner who had dedicated a lot of time, energy and passion to health and wellness,” Carney said earlier this week. “As surgeon general he will be a powerful messenger on these issues and each of the tenets of health — nutrition, activity and resilience.”

In the NRA letter sent to members, the gun-rights group said Murthy's positions in favor of some gun control measures disqualify him from the post. The letter cites Murthy's position in favor of licensing for gun and ammunition, a ban on semi-automatic weapons and a gun buy back program. It also noted his support to remove restrictions that prevent the Centers for Disease Control from "funding anti-gun advocacy research."

A New York Times article last weekend cited a number of eventual possibilities for Murthy's nomination, including delaying it until after the midterm elections or allowing him to withdraw.

Earlier this month the administration's nominee to head the Civil Rights Division of the Justice Department stalled after a number of Democrats voted against him — the first time a nomination vote has failed since the Senate amended its rules to require only a majority to pass most nominees, rather than the usual 60 votes to break a filibuster.